Partial Derivatives

Core Definition

Let $z = f(x, y)$ be a function of two variables. The first-order partial derivative of $f$ with respect to $x$ at $(x, y)$ is defined by the difference quotient limit:

\frac{\partial f}{\partial x}(x, y) = f_x(x, y) = \lim_{h \to 0} \frac{f(x + h, y) - f(x, y)}{h}

Similarly, the first-order partial derivative of $f$ with respect to $y$ at

(x, y)

is defined by:

\frac{\partial f}{\partial y}(x, y) = f_y(x, y) = \lim_{h \to 0} \frac{f(x, y + h) - f(x, y)}{h}

provided these limits exist.

Concept Explanation

The Core Operational Rule

In single-variable calculus, the derivative $\frac{df}{dx}$ tracks the instantaneous rate of change of a curve $y = f(x)$ . In multivariable calculus, the input is multidimensional: a function $z = f(x, y)$ defines a 2D surface hovering in 3D space $\mathbb{R}^3$ . Because $x$ and $y$ can vary independently and along infinite combinations of trajectories, we isolate variation along one coordinate axis at a time:

The Fundamental Rule of Partial Differentiation:

To compute $\frac{\partial f}{\partial x}$ , differentiate with respect to $x$ treating $y$ as an absolute constant coefficient.

To compute $\frac{\partial f}{\partial y}$ , differentiate with respect to $y$ treating $x$ as an absolute constant coefficient.

Holding y = b fixed:    f(x, b) is purely a function of x  ===>  d/dx[f(x, b)] = f_x(x, b)
Holding x = a fixed:    f(a, y) is purely a function of y  ===>  d/dy[f(a, y)] = f_y(a, y)

Common Notational Conventions

The curly partial symbol " $\partial$ " distinguishes partial derivatives from single-variable ordinary derivatives ( $d$ ):

With respect to $x$ : $f_x(x, y) = \frac{\partial f}{\partial x} = \frac{\partial z}{\partial x} = \partial_x f = D_1 f = D_x f$
With respect to $y$ : $f_y(x, y) = \frac{\partial f}{\partial y} = \frac{\partial z}{\partial y} = \partial_y f = D_2 f = D_y f$
Evaluated at a point $(a, b)$ : $f_x(a, b) = \left.\frac{\partial f}{\partial x}\right|_{(a, b)} = \left.\frac{\partial z}{\partial x}\right|_{(x=a, y=b)}$

Preservation of Single-Variable Calculus Rules

Because non-differentiated variables are treated as constants, every standard rule from single-variable differentiation applies directly:

Constant Multiple Rule: $\frac{\partial}{\partial x}[c(y) \cdot g(x)] = c(y) \cdot g'(x)$
Product Rule: $\frac{\partial}{\partial x}[u(x, y) \cdot v(x, y)] = \frac{\partial u}{\partial x} v + u \frac{\partial v}{\partial x}$
Quotient Rule: $\frac{\partial}{\partial x}\left[\frac{u(x, y)}{v(x, y)}\right] = \frac{\frac{\partial u}{\partial x} v - u \frac{\partial v}{\partial x}}{v^2}$
Multivariable Chain Rule (Single Outer Function): $\frac{\partial}{\partial x}[g(u(x, y))] = g'(u(x, y)) \cdot \frac{\partial u}{\partial x}$

Engineering Applications

Aerospace & Thermal Engineering: Fourier's law of heat conduction relates heat flux vector components to directional temperature gradients: $q_x = -k \frac{\partial T}{\partial x}$ .
Fluid Mechanics & Hydraulics: Viscous shear stress $\tau$ in boundary-layer pipe flow depends on the transverse velocity gradient $\frac{\partial v}{\partial y}$ .
Electrical & Computer Engineering: In transistor semiconductor physics, gate leakage current variations $\frac{\partial I_D}{\partial V_{GS}}$ define device transconductance.
Structural Mechanics: In plane stress analysis of airframes, normal strain rates $\epsilon_x = \frac{\partial u}{\partial x}$ determine elastic deformation along structural stringers.

Example Problems

Example 1: Differentiating with Polynomials and Radicals

Compute $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for $f(x, y) = x^3 y^2 - \sqrt{x^2 + y^2}$ .

Solution

Differentiate with respect to $x$ :
$\frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(x^3 y^2) - \frac{\partial}{\partial x}\left((x^2+y^2)^{1/2}\right)$
- Holding $y$ constant, $\frac{\partial}{\partial x}(x^3 y^2) = (3x^2)(y^2) = 3x^2 y^2$ .
- Using the chain rule on the radical: $\frac{\partial}{\partial x}\left((x^2+y^2)^{1/2}\right) = \frac{1}{2}(x^2+y^2)^{-1/2} \cdot \frac{\partial}{\partial x}(x^2+y^2) = \frac{1}{2\sqrt{x^2+y^2}} \cdot 2x = \frac{x}{\sqrt{x^2+y^2}}$ $\frac{\partial f}{\partial x} = 3x^2 y^2 - \frac{x}{\sqrt{x^2+y^2}}$
Differentiate with respect to $y$ :
$\frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(x^3 y^2) - \frac{\partial}{\partial y}\left((x^2+y^2)^{1/2}\right)$
- Holding $x$ constant, $\frac{\partial}{\partial y}(x^3 y^2) = (x^3)(2y) = 2x^3 y$ .
- Using the chain rule on the radical: $\frac{\partial}{\partial y}\left((x^2+y^2)^{1/2}\right) = \frac{1}{2}(x^2+y^2)^{-1/2} \cdot 2y = \frac{y}{\sqrt{x^2+y^2}}$ $\frac{\partial f}{\partial y} = 2x^3 y - \frac{y}{\sqrt{x^2+y^2}}$

Example 2: Differentiating Rational and Trigonometric Functions

Compute $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for $f(x, y) = \frac{x \sin(y)}{x^2 + 1}$ .